Complete reducibility for reductive Lie algebras

A finite-dimensional module of a reductive Lie algebra in characteristic zero is completely reducible if the center acts semisimply and its actions split over the field. In particular, over an algebraically closed field it suffices that every central element act semisimply. The condition on the center is essential: an abelian Lie algebra can act by a nonzero nilpotent operator.

The center's weight spaces are submodules for the whole Lie algebra. On each such space the center acts by scalars, so Weyl's theorem for the semisimple derived ideal supplies complements for the whole algebra. Since these weight spaces span, their sum is completely reducible. This is the representation-theoretic step that lets semisimple central actions supplement Weyl's theorem when the Lie algebra has nonzero center.

References

• J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, §§6 and 19.
• J. S. Milne, Algebraic Groups (2017), Chapter 22.
• TauCeti.Algebra.Lie.Reductive.Basic, for the center/derived-ideal decomposition, and TauCeti.Algebra.Lie.HighestWeight.CompleteReducibility, for Weyl's theorem.

theorem TauCeti.exists_isCompl_of_forall_center_lie_eq_smul {K : Type u} [Field K] [CharZero K] {L : Type v} [LieRing L] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.HasCentralRadical K L] {M : Type w} [AddCommGroup M] [Module K M] [LieRingModule L M] [LieModule K L M] [FiniteDimensional K M] (hZ : ∀ z ∈ LieAlgebra.center K L, ∃ (c : K), ∀ (m : M), ⁅z, m⁆ = c • m) (N : LieSubmodule K L M) : ∃ (P : LieSubmodule K L M), IsCompl N P

If the center acts by scalars on a finite-dimensional module of a reductive Lie algebra, every Lie submodule has a complement. The ground field need not be algebraically closed.

theorem TauCeti.complementedLattice_lieSubmodule_of_iSup_weightSpace_center_eq_top {K : Type u} [Field K] [CharZero K] {L : Type v} [LieRing L] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.HasCentralRadical K L] {M : Type w} [AddCommGroup M] [Module K M] [LieRingModule L M] [LieModule K L M] [FiniteDimensional K M] (hZ : ⨆ (χ : ↥(LieAlgebra.center K L) → K), ↑(LieModule.weightSpace M χ) = ⊤) : ComplementedLattice (LieSubmodule K L M)

A finite-dimensional module of a reductive Lie algebra is completely reducible whenever its honest central weight spaces span. This split form works over the original field.

theorem TauCeti.complementedLattice_lieSubmodule_of_isSemisimple_center {K : Type u} [Field K] [CharZero K] {L : Type v} [LieRing L] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.HasCentralRadical K L] {M : Type w} [AddCommGroup M] [Module K M] [LieRingModule L M] [LieModule K L M] [FiniteDimensional K M] [LieModule.IsTriangularizable K (↥(LieAlgebra.center K L)) M] (hZ : ∀ (z : ↥(LieAlgebra.center K L)), ((LieModule.toEnd K (↥(LieAlgebra.center K L)) M) z).IsSemisimple) : ComplementedLattice (LieSubmodule K L M)

Over a splitting field for the central action, semisimplicity of each central operator implies complete reducibility of a finite-dimensional module of a reductive Lie algebra.